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## Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top

Regular and Chaotic Dynamics. 2016. Vol. 21. No. 7-8. P. 885-901.

This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of period-doubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.

Keywords: bifurcationstrange attractornonholonomic constraintChaplygin Toprubber modeltrajectory of the point of contact

Publication based on the results of:

Kazakov A., Борисов А. В., Пивоварова Е. Н., Нелинейная динамика 2017 Т. 13 № 2 С. 277-297

This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario ...

Added: October 13, 2017

Bizyaev I. A., Borisov A. V., Kazakov A., Regular and Chaotic Dynamics 2015 Vol. 20 No. 5 P. 605-626

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the ...

Added: October 22, 2015

Beklaryan L. A., Beklaryan A., Computational Mathematics and Mathematical Physics 2020 Vol. 60 No. 8 P. 1249-1260

The importance of functional differential equations of pointwise type is determined by the fact that their solutions are used to construct traveling-wave solutions for induced infinite-dimensional ordinary differential equations, and vice versa. Solutions of such equations exhibit bifurcation. A theorem on branching bifurcation is obtained for the solution to a linear homogeneous functional differential equation of pointwise type. ...

Added: September 21, 2020

Гонченко С. В., Исаенкова Н. В., Zhuzhoma E. V., Журнал Средневолжского математического общества 2013 Т. 15 № 1 С. 76-79

Приводятся бифуркации разрушения и рождения соленоидов Смейла-Вильямса ...

Added: October 17, 2014

Kazakov A., Korotkov A., Osipov G. V., Regular and Chaotic Dynamics 2015 Vol. 20 No. 6 P. 701-715

In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of generalized Lotka-Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of ...

Added: October 22, 2015

Kazakov A., Баханова Ю. В., Коротков А. Г., Журнал Средневолжского математического общества 2017 Т. 19 № 2 С. 13-24

Investigations of spiral chaos in generalized Lotka-Volterra systems and Rosenzweig-MacArthur systems that describe the interaction of three species are made in this work. It is shown that in systems under study the spiral chaos appears in agreement with Shilnikov's scenario, that is when changing a parameter in system a stable limit cycle and a saddle-focus ...

Added: October 13, 2017

Korotkov A., Kazakov A., Леванова Т. А. et al., Communications in Nonlinear Science and Numerical Simulation 2019 Vol. 71 P. 38-49

We investigated the phenomenological model of ensemble of two FitzHugh–Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model the coupling which is implemented by smooth function that approximates rectangular function and reflects main important properties of biological synaptic coupling. The proposed coupling depends on three ...

Added: October 18, 2019

Kuryzhov E., Karatetskaia E., Mints D., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 2 P. 165-174

We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble ...

Added: September 8, 2021

Shalimova E., Burov A. A., Technische Mechanik 2017 Vol. 37 No. 2-5 P. 129-138

Dynamics of a massive point on a rotating wire or surface under dry friction force action is considered. Existence, stability and bifurcations of non-isolated relative equilibria sets of the point located - on a sphere uniformly rotating about an inclined fixed axis; - on a thin circular hoop rotating about an inclined fixed axis; - ...

Added: December 7, 2017

Gonchenko S. V., Gonchenko A. S., Kazakov A. et al., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2018 Vol. 28 No. 11 P. 1830036-1-1830036-29

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the ...

Added: October 26, 2018

Борисов А. В., Kazakov A., Сатаев И. Р., Regular and Chaotic Dynamics 2016 Vol. 21 No. 7-8 P. 939-954

This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that ...

Added: November 21, 2016

Бизяев И. А., Борисов А. В., Kazakov A., Нелинейная динамика 2016 Т. 12 № 2 С. 263-287

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a ﬁxed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-ﬁxed axis is equal to zero. Depending on the ...

Added: October 29, 2016

Kazakov A., Козлов А. Д., Журнал Средневолжского математического общества 2018 Т. 20 № 2 С. 187-198

In the paper a new method of constructing of three-dimensional flow systems with different chaotic attractors is presented. Using this method, an example of three-dimensional system possessing an asymmetric Lorenz attractor is obtained. Unlike the classical Lorenz attractor, the observed attractor does not have symmetry. However, the discovered asymmetric attractor, as well as classical one, ...

Added: October 26, 2018

Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane

Kazakov A., Regular and Chaotic Dynamics 2013 Vol. 18 No. 5 P. 508-520

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The termrubbermeans that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose ...

Added: March 29, 2015

Kazakov A., Borisov A. V., Sataev I. R., Regular and Chaotic Dynamics 2014 Vol. 19 No. 6 P. 718-733

In this paper we consider the motion of a dynamically asymmetric unbalanced ball
on a plane in a gravitational field. The point of contact of the ball with the plane is subject
to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the
nonholonomic reversible system of 6 differential equations. In the case ...

Added: March 29, 2015

Kazakov A., Борисов А. В., Кузнецов С. П., Успехи физических наук 2014 Т. 184 № 5 С. 493-500

Based on the results of numerical simulations we discuss and illustrate dynamical phenomena characteristic for the rattleback, a solid body of convex surface moving on a rough horizontal plane, which are associated with the lack of conservation for the phase volume in the nonholonomic mechanical system. Due to local compression of the phase volume, behaviors ...

Added: October 22, 2015

Kulagin N., Lerman L., Malkin A., Communications in Nonlinear Science and Numerical Simulation 2021 Vol. 93 Article 105525

Solitons and cavitons (the latter are localized solutions with singularities) for the nonlocal Whitham equations are studied. The fourth order differential equation for traveling waves with a parameter in front of the fourth derivative is reduced to a reversible Hamiltonian system defined on a two-sheeted four-dimensional space. Solutions of the system which stay on one ...

Added: September 16, 2020

Kazakov A., Korotkov A., Levanova T. et al., IFAC-PapersOnLine 2018

We study the peculiarities of chaotic dynamics in the phenomenological model of the ensemble of two FitzHugh-Nagumo elements with weak excitatory couplings. This model was recently proposed as a suitable model for describing the behaviour of two coupled neurons. A rich diversity of different types of neuron-like behaviour, including regular in-phase, anti-phase, sequential spiking activities ...

Added: October 26, 2018

Kazakov A., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 8-9 С. 729-738

In this paper, a new scenario of the appearance of mixed dynamics in two-dimensional reversible diffeomorphisms is proposed. The key point of the scenario is a sharp increase of the sizes of both strange attractor and strange repeller which appears due to heteroclinic bifurcations of the invariant manifolds of saddle fixed points belonging to these ...

Added: October 26, 2018

Kazakov A., Gonchenko A. S., Gonchenko S. V. et al., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2014 Vol. 24 No. 8 P. 1440005-1440030

We give a qualitative description of two main routes to chaos in three-dimensional maps. We discuss Shilnikov scenario of transition to spiral chaos and a scenario of transition to discrete Lorenz-like and figure-eight strange attractors. The theory is illustrated by numerical analysis of three-dimensional Henon-like maps and Poincar´ e maps in models of nonholonomic mechanics ...

Added: March 29, 2015

Kazakov A., Гонченко С. В., Гонченко А. С. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2017 Т. 25 № 2 С. 4-36

We consider important problems of modern theory of dynamical chaos and its applications. At present, it is customary to assume that in the finite-dimensional smooth dynamical systems three fundamentally different forms of chaos can be observed. This is the dissipative chaos, whose mathematical image is a strange attractor; the conservative chaos, for which the whole ...

Added: October 13, 2017

Бекларян Л. А., Beklaryan A., Журнал вычислительной математики и математической физики 2020 Т. 60 № 8 С. 1291-1303

The importance of functional differential equations of pointwise type is determined by the fact that their solutions are used to construct traveling-wave solutions for induced infinite-dimensional ordinary differential equations, and vice versa. Solutions of such equations exhibit bifurcation. A theorem on branching bifurcation is obtained for the solution to a linear homogeneous functional differential equation ...

Added: August 25, 2020

Goncharuk N. B., Ilyashenko Y., Солодовников Н. А., Moscow Mathematical Journal 2019 Vol. 19 No. 4 P. 709-737

We classify global bifurcations in generic one-parameter local families of vector elds on S2 with a parabolic cycle. The classication is quite dierent from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable. ...

Added: October 23, 2020

Kazakov A., Гонченко А. С., Гонченко С. В. et al., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 10 С. 867-882

We study dynamical properties of a Celtic stone moving along the plane. Both one- and two-parameter families of the corresponding nonholonomic models are considered, in which bifurcations are studied that lead to changing types of stable motions of the stone as well as to the onset of chaotic dynamics. It is shown that multistability phenomena ...

Added: October 26, 2018